Bounds for KdV and the 1-d cubic NLS equation in rough function spaces

Koch, Herbert

Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 11, 10 p.

We consider the cubic Nonlinear Schrödinger Equation (NLS) and the Korteweg-de Vries equation in one space dimension. We prove that the solutions of NLS satisfy a-priori local in time $H^s$ bounds in terms of the $H^s$ size of the initial data for $s\ge -\frac{1}{4}$ (joint work with D. Tataru, [15, 14]) , and the solutions to KdV satisfy global a priori estimate in $H^{-1}$ (joint work with T. Buckmaster [2]).

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